The object of the course is to familiarize the students with the basic language of and some fundamental theorems in the theory of real valued functions of one rela variable, focusing on fundamental notions as sequences, continuos functions, derivatives, integrals .
Course contents summary
1) Real numbers and functions of one real variable.
Axiomatic theory of real numbers.
Natural, Integres and Ratinal numbers.
Power, trigonometric, exponential and logarithmic functions .
Maximum, minimum, infimum and supremum.
Sequences: definition and examples.
Limit of sequences.
3) Continuos functions.
Functions and limits.
Continuos functions: examples and basic properties.
Limit of functions and limit of sequences.
Derivatives: definition and examples.
Geometrical meaning of derivatives: tangent lines.
Rules for calculation of derivatives.
Derivatives of elementary functions.
Derivatives of higher order.
5) Fundamental Theorems of differentiable functions.
Rolle, Lagrange and Cauchy Theorems and consequences.
Extremal points. Local maximum and local minimum of functions.
6) Theory of Riemann integration.
Notation. The Riemann integral.
Basic properties of Riemann integrable functions.
Definite integrals: geometrical interpretation.
Mean Value Theorems for Integrals.
Fundamental theorem of calculus. Fundamental formula of calculus. indefinite integrals. Methods of integrations.
P. Marcellini, C. Sbordone: Calcolo, Liguori Editore, P. Marcellini, C. Sbordone: Esercitazioni di matematica, Volume 1, parte prima e seconda, Liguori Editore.
Theoretical lectures and sessions of oral and written exercises.